> and
is a smooth function.
A real hypersurface is said to be a Hopf hypersurface if
is principal.
The Jacobi operator field with respect to X on M is de-
fined by
,RXX
=X
, where R is the Riemmanian curva-
ture of M. For
the Jacobi operator is called
structure Jacobi operator and is denoted by
=,lR
.
It has a fundamental role in almost contact manifolds.
Many differential geometers have studied real hypersur-
faces in terms of the structure Jacobi operator.
The Lie derivative of the structure Jacobi operator
with respect to
was investigated by Perez, Santos,
Suh (see [7]). More precisely, they classified real hyper-
surfaces in n
P
3
=0l
2
(n), whose structure Jacobi op-
erator satisfies the condition:
L. Ivey and Ryan
(see [8]) classified real hypersurfaces satisfying the same
condition in
P
2
and
H
.
The study of real hypersurfaces whose structure Jacobi
operator is parallel is a problem of great importance. In
[9] the nonexistence of real hypersurfaces in nonflat
C
opyright © 2012 SciRes. APM
K. PANAGIOTIDOU ET AL.
2


,= ,,
,= ,
complex space form with parallel structure Jacobi opera-
tor () was proved. In [10] a weaker condition
(-parallelness), X for any vector field X or-
thogonal to
g
XY gXYXY
gX YgXY
 

 
=,
=,
X
X
AX
YYAXgAXY
=0l
=0l
, was studied and it was proved the non-
existence of such hypersurfaces in case of n
P
3
n
(n).
The parallelness of structure Jacobi operator in combina-
tion with other conditions was another problem that was
studied by many others such as Ki, Kim, Perez, Santos,
Suh (see [11,12]).
Recently Perez-Santos (see [1]) studied real hypersur-
faces in
P
for , whose structure Jacobi opera-
tor satisfies the relation:
3n
=.ll

L
P H
P
H
(1)
In the present paper we go on studying the same prob-
lem for 2
and 2. We prove the following theo-
rem:
Main Theorem Let M be a real hypersurface in 2
or 2, whose structure Jacobi operator satisfies rela-
tion (1). Then M is locally congruent to: a geodesic
sphere of radius r, where π
<2
r0< with π
4
r, or to
a tube of radius π
=4
r over a holomorphic curve in
2 and to a horosphere, a geodesic sphere or a tube
over
P
1
H
A
in 2i or to a Hopf hypersurface in 2
with
HH
=0
.
2. Preliminaries
Let M be a connected real hypersurface immersed in a
nonflat complex space form

,
n
M
cG 0c
=
, (), with
almost complex structure J of constant holomorphic sec-
tional curvature c. Let N be a unit normal vector field on
M and
J
N
. For a vector field X tangent to M we
can write
 
=
J
XX

XN
, where φX and
X
N
are the tangential and the normal component of JX re-
spectively. The Riemannian connection
in
n
M
c
and in M are related for any vector fields X, Y on M:
=,
YY
X
XgAYXN
=
XNAX
where g is the Riemannian metric on M induced from G
of
n
M
c and A is the shape operator of M in
n
M
c
(,,)
.
M has an almost contact metric structure

in-
duced from J on
n
M
c where
is a (1,1) tensor field
and
is a 1-form on M such that

,= ,,
g
XY
G JXY
 
= ,

=,
X
gX

GJXN
 
=0,X

(see [13]). Then we have


2=,
=0, =1
XX X

 
 (2)
(3)



(4)
Since the ambient space is of constant holomorphic
sectional curvature c, the equations of Gauss and Codazzi
for any vector fields X, Y, Z on M are respectively given
by
 
 
 
,= ,,,
4
,2,
,,
c
RXYZgYZXgXZY gYZX
gXZYgXYZ
gAYZAXgAXZAY
 

 


(5)
 

=4
2,
XY
c
A
YAX XYYX
gXY


 

PM
(6)
where R denotes the Riemannian curvature tensor on M.
For every point
, the tangent space can
be decomposed as following:
P
TM

=
P
T Mspanker
where

=:=0kerXTMX

P. Due to the
above decomposition, the vector field
A
is decom-
posed as follows:
A
U=
 
(7)
where =

and

1
=U ker


0
, pro-
vided that
.
All manifolds are assumed connected and all mani-
folds, vector fields etc are assumed smooth (C
).
3. Auxiliary Relations
2
P
2
or Suppose now that the ambient space is
H
,
(i.e.
2c0c,
M
), then we consider V be the open
subset of points P
M, such that there exists a neighbor-
hood of every P, where =0
and the open subset
of points Q of M such that there exists a neighborhood of
every Q, where
0a
. Since,
is a smooth function
on M, then is an open and dense subset of M. V
Proposition 3.1 Let M be a real hypersurface in
2
M
c, equipped with structure Jacobi operator satis-
fying (1). Then,
is principal on V.
A
Proof: The relation (7) on V, takes the form
U
X
UYZ
. From (5) for
,

we obtain:
.
4
c
lU U (8)
Copyright © 2012 SciRes. APM
K. PANAGIOTIDOU ET AL. 3
Due to the definition of Lie derivative, the relation (1)
for X
yields: l
0

0
. The latter, because of the
first relation of (4) and (8) implies
A
, hence
0
. Therefore,
is principal on V.
On if
=0
, then
is principal. In what fol-
lows we work on W (W), which is the open subset
of points such that

0
Q
.
Lemma 3.2 Let M be a real hypersurface in
2
M
c.
Then the following relations hold on W:
2
,
44
cc
A
UU
AUU
 





 (9)
2
,
44
UU
cc
UU
,
U

 

 

 (10)
12
,,
4
UU
c
UUU U
 
 3
UU


(11)
2
23
,
UU
14
,
U
U
c
UU
UU

 





3
,,

,,UU


 
(12)
where are smooth functions on M.
12
Proof: If
=,=,
is an orthonormal basis, then
because of (7) we have:
A
UU UAUUU
 
 (13)
where
,
and
are smooth functions on M.
The first relation of (4), because of (13) yields:
=,
=.
U
U
= ,
U
UU UU


 (14)
The relation (5), using (13) can be written:
2
4
c
lU
lU U
,
.
4
U U
cU











.
lX X
l
(15)
By the definition of Lie derivative, the relation (1)
takes the form:


The latter for X
,U
0lU implies lU = 0,
and then from (15) we obtain:
2
44
cc
 
0, ,.

 

,
XX
(16)
The relations (13) and (14), because of (16) imply (9)
and (10) respectively.
From the well known relation:
,,
X
gYZgYZ
,,
gY Z
for
X
YZ
,,UU
, using (16) we obtain (11) and
(12), where 1
, 2
and 3
are smooth functions on M.

,,UU

The Codazzi equation for X, Y , because
of Lemma 3.2 yields:
2
2
41,Uc


 


(17)
22
3
1,
44
cc

 

 


(18)
2
2
4,Uc


 (19)
2
2
4,
c




(20)
3
3,
4
c
U


 



(21)
2
2
1,
24
cc
U



 



(22)
22
13.
44
cc
U




 
 
 
 
(23)
The Riemannian curvature on M satisfies (5) and on
the other hand is given by the relation
,RXYZZZ Z

,,UU



,
XY YXXY . From these two
relations and because of (16) for X, Y we
obtain:
2
3
21
2
22
3
12
242
c
cc
UU
c

 
 

 


(24)

2
312 3
4
c
U

 


 



(25)
3213 3
42
cc
U


 


34
(26)
Relation (23), because of (18), (21) and (22), yields:
(27)
and so relation (18) becomes:
2
2
14.
44
cc





(28)
Differentiating the relations (27) and (28) with respect
to U and
respectively and substituting in (25) and
due to (19), (20) and (27) we obtain:
22
224 0c

. (29)
Copyright © 2012 SciRes. APM
K. PANAGIOTIDOU ET AL.
4
Owing to (29), we consider , (W) the open
subset of points , where in a neighbor-
hood of every Q.
1
W1W
20
QW
Lemma 3.3 Let M be a real hypersurface in
2
M
c
1
W
2
c
,
equipped with Jacobi operator satisfying (1). Then
is empty.
Proof: Due to (29) we obtain: on 1
W.
Differentiation of the last relation along
2
24

and taking
into account (19), (20) and yields: ,
which is a contradiction. Therefore, is empty.
2
24
2
c

=0c
2=0
0.
1
On W, because of Lemma 3.3, we have , hence
the relations (17), (19) and (20) become:
W
UU




Using the last relations and Lemma 3.2 we obtain:
,0,U
 
UU
 




.cU
22
,
1416
4
U
UU
 



0cU


WWQW
 

Combining the last two relations we have:
22
416

 . (30)
Let 2, (2) be the set of points W
, for
which there exists a neighborhood of every Q such that
. So from (30) we have: 16 .
Differentiating the last relation with respect to

U

0 22
4c


U
and
taking into account (21), (22), (27) and (28), we have:
, which is impossible. So 2 is empty. Hence,
on W we have . Then, relations (21) and
(28), because of (27) imply respectively:
20
W

U

0
2
4c
2
and
2
5
1
 
 . Relation (26), because of

0U

,
2
4c
and (27) yields: 1
2
. Substitution of 1
in 1
2
52


yields: 322
. Differentiation of
the last one along U
and taking into account (22) leads
to: 0
, which is a contradiction. So we obtain the
following proposition:
Proposition 3.4 Let M be a real hypersurface in
2

M
c, equipped with Jacobi operator satisfying (1).
Then M is a Hopf hypersurface.
4. Proof of Main Theorem
Since M is a Hopf hypersurface, we can write
A
ZZ
and
A
ZZ

with
,,
Z
Z
, being a local or-
thonormal basis and the following relation holds:

24
c
 


(Corollary 2.3 [14]). Furthermore,
due to Theorem 2.1 [14], we have that
is constant.
From (5), due to
A
ZZ
and
A
ZZ

we
get:
,.
44
cc
lZZl ZZ

 
 
 
 
(31)
The relation (1), because of (31) implies:
0
 
, for
X
Z
and , for

0
 

X
Z
. Combining the last two relations leads to:

20

2
P
, M is locally
0
. If
, in case of
congruent to a tube of radius π
4
2
over a holomorphic
P
curve in
0A
due to [15] and to a Hopf hypersurface
with 2
H
. If
in 0
, the last relation im-
plies:
and we obtain:
0, ,
A
AX XTM


which because of Theorem A completes the proof of
Main Theorem.
5. Acknowledgements
The first author is granted by Foundation Alexandros S.
Onasis. Grant Nr: G ZF 044/2009-2010. The authors
would like to thank Professors Juan de Dios Perez and
Patrick J. Ryan. Their answers improved the proof of
main theorem. Finally, the authors express their grati-
tude to the referee who has contributed to improve the
paper.
REFERENCES
[1] J. D. Perez and F. G. Santos, “Real Hypersurfaces in
Complex Projective Space Whose Structure Jacobi Op-
erator Satisfies
R
R

,” Rocky Mountain Journal
of Mathematics, Vol. 39, No. 4, 2009, pp. 1293-1301.
L
doi:10.1216/RMJ-2009-39-4-1293
[2] R. Takagi, “On Homogeneous Real Hypersurfaces in a
Complex Projective Space,” Osaka Journal of Mathe-
matics, Vol. 10, 1973, pp. 495-506.
[3] J. Berndt, “Real Hypersurfaces with Constant Principal
Curvatures in Complex Hyperbolic Space,” Journal fur
die Reine und Angewandte Mathematik, Vol. 395, 1989,
pp. 132-141. doi:10.1515/crll.1989.395.132
[4] M. Okumura, “On Some Real Hypersurfaces of a Com-
plex Projective Space,” Transactions of the American
Mathematical Society, Vol. 212, 1975, pp. 355-364.
doi:10.1090/S0002-9947-1975-0377787-X
[5] S. Montiel and A. Romero, “On Some Real Hypersur-
faces of a Complex Hyperbolic Space,” Geometriae Dedi-
cata, Vol. 20, No. 2, 1986, pp. 245-261.
doi:10.1007/BF00164402
[6] U.-H. Ki and H. Liu, “Some Characterizations of Real
Hypersurfaces of Type (A) in Nonflat Complex Space
Form,” KMSBulletin of the Korean Mathematical Society,
Vol. 44, No. 1, 2007, pp. 157-172.
doi:10.4134/BKMS.2007.44.1.157
[7] J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersur-
Copyright © 2012 SciRes. APM
K. PANAGIOTIDOU ET AL.
Copyright © 2012 SciRes. APM
5
faces in Complex Projective Space Whose Structure Ja-
cobi Operator Is Lie
-Parallel,” Differential Geometry
and Its Applications, Vol. 22, No. 2, 2005, pp. 181-188.
doi:10.1016/j.difgeo.2004.10.005
[8] T. A. Ivey and P. J. Ryan, “The Structure Jacobi Operator
for Real Hypersurfaces in 2
P and 2
H,” Results in
Mathematics, Vol. 56, No. 1-4, 2009, pp. 473-488.
doi:10.1007/s00025-009-0380-2
[9] M. Ortega, J. D. Perez and F. G. Santos, “Non-Existence
of Real Hypersurfaces with Parallel Structure Jacobi Op-
erator in Nonflat Complex Space Forms,” Rocky Moun-
tain Journal of Mathematics, Vol. 36, No. 5, 2006, pp.
1603-1613. doi:10.1216/rmjm/1181069385
[10] J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersur-
faces in Complex Projective Space Whose Structure Jacobi
Operator Is D-Parallel,” Bulletin of the Belgian Mathe-
matical Society Simon Stevin, Vol. 13, No. 3, 2006, pp.
459-469.
[11] U.-H. Ki, J. D. Perez, F. G. Santos and Y. J. Suh, “Real
Hypersurfaces in Complex Space Forms with ξ-Parallel
Ricci Tensor and Structure Jacobi Operator,” Journal of
the Korean Mathematical Society, Vol. 44, No. 2, 2007,
pp. 307-326. doi:10.4134/JKMS.2007.44.2.307
[12] N.-G. Kim and U.-H. Ki, “ξ-Parallel Structure Jacobi
Operators of Real Hypersurfaces in a Nonflat Complex
Space Form,” Honam Mathematical Journal, Vol. 28, No.
4, 2006, pp. 573-589.
[13] D. E. Blair, “Riemannian Geometry of Contact and Sym-
plectic Manifolds,” Progress in Mathematics 203, Birk-
häuser, Boston, 2002.
[14] R. Niebergall and P. J. Ryan, “Real Hypersurfaces in
Complex Space Forms,” Publications of the Research In-
stitute for Mathematical Sciences, Vol. 32, 1997, pp. 233-
305.
[15] T. Cecil and P. J. Ryan, “Focal Sets and Real Hypersur-
faces in Complex Projective Spaces,” Transactions of the
American Mathematical Society, Vol. 269, No. 2, 1982,
pp. 481-499.